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Valdez, the former Dallas County sheriff, has been under fire for being too accommodating of federal immigration authorities. White jumped in, criticizing her jail management record as subpar when it came to handling people who were in the country illegally.

He lavished praise on Travis County Sheriff Sally Hernandez in Austin, who is also a Democrat, saying, "She did not work with ICE the way that Sheriff Valdez worked with ICE."

"I did not work with ICE," she said. "I did what I had to do, and that was an imperfect choice. What I did do ... is make sure that the people who were brought in received humane services."

Valdez quickly counterpunched on abortion, an issue on which she said White has shown insensitivity.

"Andrew, you've implied that women that have an abortion do not respect life," she said. "You owe an apology to these women."

White fired back, "I have not implied that."

White, who is an elder in a conservative Presbyterian church in Houston, has said that he personally opposes abortion but that as governor would veto bills to further restrict it.

"My personal opinions are my personal opinions," he said. "As governor, I trust women to make their own health care decisions."

Valdez played down any damage that her own strong support for abortion rights might do to her prospects of attracting fellow Hispanics' support. An internet ad and website launched by GOP Gov. Greg Abbott's re-election campaign had raised that likelihood earlier this week.

"I don't think a wedge will come between a person who's trying to represent the average, everyday Texan," she said. "We will be able to get that message out to all the Hispanics."

Valdez, the first Hispanic lesbian elected sheriff in Texas, also bristled when White bragged about winning the endorsement of the Houston GLBT Political Caucus.

Valdez shot back: "I did get the largest LGBT organization in Texas. Texas Equality endorsed me."

If Valdez becomes the Democratic nominee, Abbott and the Republicans are almost sure to remind voters of some of her debate remarks.

"Abortion is health care," she said at one point.

Also, though she ruled out a state income tax, she seemed open to ending the exemption of grocery purchases from the state's 6.25-cent sales tax.

"We have to re-examine and decide which ones we need to get rid of," she said of sales tax exemptions.

Asked how to reduce the numbers of African-Americans and Hispanics who are in prison, Valdez replied, "We have to invest in those areas so that the automatic thing is not to turn to crime."

She said the first thing she would do is put minorities in management positions.

"The community does not trust law enforcement that does not look like them," she said.

White said decriminalizing possession of small amounts of marijuana is key to reducing mass incarceration.

Bail bond laws need to be changed, he added, as they impose hardships on the poor.

White hit Abbott for recently deploying the National Guard to the Texas-Mexico border. He said it was showboating.

In discussing his top priority as governor, White spoke of education.

"If I have to activate the National Guard to teach kids how to read, I'll do it," he said.

He also criticized Abbott for calling a special legislative session for a bathroom bill but not after Hurricane Harvey. "It's disgusting," White said, "and I'll never forgive him for it."

The Democratic rivals agreed that President Donald Trump's proposals to have some teachers carry guns are wrongheaded. Both also denounced a lack of affordable housing in Texas' biggest cities, as poor neighborhoods are gentrified.

White and Valdez also both said Texas needs universal prekindergarten, though neither supports higher taxes.

White, a businessman and son of the late Gov. Mark White, said he has offered a plan for stopping erosion of local property tax receipts -- by stopping downward revisions of commercial property appraisals.

That would help pay for some of his education plans, he said, though the move wouldn't directly enable the Legislature to spend more money.

Abbott has been in statewide office since 1995. And Jeffers noted that the state's major newspapers have endorsed White, citing concerns about Valdez's grasp of state issues.

"It's not that I'm not as sharp" as White, Valdez responded. "The problem may be that I don't talk newspaper language. I talk people language."

Speaking with reporters afterward, though, White said Valdez "didn't offer any new ideas." He also urged people to review Valdez's Facebook Live appearance before the Fort Worth Star-Telegram's editorial board earlier this week.

"It's a 30-minute spectacle," said White, who also appeared before the board.

White said he could raise more money than Valdez to compete with Abbott, though neither Democratic hopeful would tell reporters how much they have raised and spent since the March 6 primary.

As for what happens after their runoff, each promised to back the other in the fall.

The one-hour debate, sponsored by a "grassroots coalition" of Democratic groups, was held before more than 200 people at St. James Episcopal Church in East Austin. It was aired on the cable TV provider Spectrum in Austin and San Antonio. Austin's KXAN-TV carried a livestream, which was shared on many news outlets' websites.

In the March primary, Valdez led a field of nine candidates. Of slightly more than 1 million votes cast, she garnered nearly 439,000. White finished second, with about 280,000 votes.

In his GOP primary against two little-known rivals, Abbott commanded 1.4 million votes, or 90 percent of those cast.

With a month of primaries left, the record for the number of female nominees for governor has already been broken. Some of them are also the first black, Latina and Native American women nominated by a major party.